If the vertical is way more pronounced than the horizontal,
we have a very steep hypotenuse (either up or down), whereas
a strong horizontal (vs. a smallish vertical) looks like a
very low grade hill (something you could easily skateboard
up or down).
So then we're talking vertical "versus" horizontal, which
I connect to y/x and y:x notation. But you can also use
the variables v/h (for vertical vs. horizontal). Since we
don't have an xy graph in the picture (yet) it's not a
problem to use v:h instead of y:x.
A useful side trip here is to point out that if you know it's
a right triangle, then one additional angle is going to tell
you everything about its shape, i.e. the h:v ratio is going
to be 100% definite given one of the acute angles. This
means we can link to the TAN function on the calculator.
I explain that TAN takes our angle as input and spits out
the numeric value of h:v (y:x).
Then I have students tell me TAN(45 deg) without using a
calculator -- because a 45 deg angle clearly keeps h:v in
a 1:1 ratio, i.e. TAN(45 deg) must be 1.
We then review the concept of "inverse function": if you
know that TAN(angle)=h/v, then ATAN(h/v) = angle. So if
you know h and v, you can tell me the angle between
opposite (v) and adjacent (h) legs. So if v is much
bigger than h, say the ratio is 340/12, then we should
expect a pretty big angle (steep slope). ATAN(340/12)
is about 88 degrees (pretty steep for sure!).
In the case of y = mx + b, you might point out that if m
is large, then a small change in x is going to result in
a big change in y, whereas if m=1, then y will change
in direct 1:1 proportion with x. So m=1 corresponds to
TAN(45 degrees) and "big m" means "tiny h increment
makes for big vertical leap... means steep!" (as in
340/12).
The only thing missing from the above presentation is
sign. By convention (because we use left-handed XYZ),
a left-to-right loss of altitude is considered
negative (plane coming in for a landing) whereas a
left-to-right gain in altitude (plane going out) is
considered positive.
Useful to invent a "cockpit instrument" for the dashboard
of your plane. Do a semi-circle from the unit circle,
with a "green zone" showing an attainable/acceptable
wedge of inclines and declines corresponding to real
gain and loss of altitude limits. Of course stunt pilots
can do vertical dives and some military jets do almost
vertical climbs, so you can say this is for a standard
commercial airliner. Then you have an indicator which
gives the slope, pointing from the origin to some point
on the rim of the semi-circle. "Going down" is "below
the horizontal" whereas "going up" is above the
horizontal.
For those interested, I've updated "Mr. Urner's Chalkboard"
(a web page of my reformed-calculus squiggles), with a new
GIF (upper left) showing some of this thinking (e.g. a
"delta airlines" cockpit instrument for safe TAN(a)).
See: http://www.inetarena.com/~pdx4d/ocn/chalkboard.html
(best viewed at 800 x 600).
Kirby
Curriculum writer
Oregon Curriculum Network
http://www.inetarena.com/~pdx4d/ocn/
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Kirby T. Urner http://www.teleport.com/~pdx4d/kirby.html
4D Solutions http://www.teleport.com/~pdx4d/ [PGP OK]
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